Related papers: Classical multiple orthogonal polynomials for arbi…
We are concerned with the monic orthogonal polynomials with respect to a singularly perturbed Laguerre-type weight. By using the ladder operator approach, we derive a complicated system of nonlinear second-order difference equations…
In this paper we consider a large class of many-variable polynomials which contains generalisations of the classical Hermite, Laguerre, Jacobi and Bessel polynomials as special cases, and which occur as the polynomial part in the…
We give integral representations for multiple Hermite and multiple Hermite polynomials of both type I and II. We also show how these are connected with double integral representations of certain kernels from random matrix theory.
Laguerre and Laguerre-type polynomials are orthogonal polynomials on the interval $[0,\infty)$ with respect to a weight function of the form $w(x) = x^{\alpha} e^{-Q(x)}, Q(x) = \sum_{k=0}^m q_k x^k, \alpha > -1, q_m > 0$. The classical…
The list of physically motivated urn models that can be solved in terms of classical orthogonal polynomials is very small. It includes a model proposed by D. Bernoulli and further analyzed by S. Laplace and a model proposed by P. and T.…
Limiting cases are studied of the Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials. We recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables…
In this paper we propose a way to construct classical type Sobolev orthogonal polynomials. We consider two families of hypergeometric polynomials: ${}_2 F_2(-n,1;q,r;x)$ and ${}_3 F_2(-n,n-1+a+b,1;a,c;x)$ ($a,b,c,q,r>0$, $n=0,1,...$), which…
Two families (type $A$ and type $B$) of confluent hypergeometric polynomials in several variables are studied. We describe the orthogonality properties, differential equations, and Pieri type recurrence formulas for these families. In the…
Multivariable generalizations of the classical Hermite, Laguerre and Jacobi polynomials occur as the polynomial part of the eigenfunctions of certain Schr\"odinger operators for Calogero-Sutherland-type quantum systems. For the generalized…
The Fisher information of the classical orthogonal polynomials with respect to a parameter is introduced, its interest justified and its explicit expression for the Jacobi, Laguerre, Gegenbauer and Grosjean polynomials found.
We wish to investigate the $D_{\omega}$-classical orthogonal polynomials, where $D_{\omega}$ is a special case of the Hahn operator. For this purpose, we consider the problem of finding all sequences of orthogonal polynomials such that…
Complementary polynomials of Legendre polynomials are briefly presented, as well as those for the confluent and hypergeometric functions, relativistic Hermite polynomials and corresponding new pre-Laguerre polynomials. The generating…
We introduce a class of orthogonal polynomials in two variables which generalizes the disc polynomials and the 2-$D$ Hermite polynomials. We identify certain interesting members of this class including a one variable generalization of the…
We introduce a new multivariate orthogonal polynomial which is a 2-parameter deformation of the spherical polynomial by harmonic analysis on symmetric cone. This is also regarded as a multivariate analogue of the circular Jacobi polynomial.…
We give four examples of families of orthogonal polynomials for which the coefficients in the recurrence relation satisfy a discrete Painlev\'e equation. The first example deals with Freud weights $|x|^\rho \exp(-|x|^m)$ on the real line,…
In this paper, we obtain the ladder operators and associated compatibility conditions for the type I and the type II multiple orthogonal polynomials. These ladder equations extend known results for orthogonal polynomials and can be used to…
Multiple orthogonal polynomials satisfy a number of recurrence relations, in particular there is a $(r+2)$-term recurrence relation connecting the type II multiple orthogonal polynomials near the diagonal (the so-called step-line recurrence…
We investigate type I multiple orthogonal polynomials on $r$ intervals which have a common point at the origin and endpoints at the $r$ roots of unity $\omega^j$, $j=0,1,\ldots,r-1$, with $\omega = \exp(2\pi i/r)$. We use the weight…
In this work we show how to get advantage from the Riemann--Hilbert analysis in order to obtain first and second order differential equations for the orthogonal polynomials and associated functions with a weight on the unit circle. We…
Mehler-Heine asymptotics describe the behavior of orthogonal polynomials near the edges of the interval where the orthogonality measure is supported. For Jacobi polynomials and Laguerre polynomials this asymptotic behavior near the hard…